Sampling error correlated among observations: Origin,impacts,and solutions

Abstract

Geoscientific datasets can contain individual data for more than 50 different chemical elements. The association between these variables is as important as their individual values. However, it is commonly overlooked that the observed covariance may be overestimated due to correlated errors. Dependent errors arise from many sources, such as the segregation process of minerals associated with these variables during delimitation, extraction, and preparation steps. This study extends a classical model composed of grade-independent (additive) and grade-proportional (multiplicative) errors to a generalised multivariate model that can estimate the real variance, covariance, and correlation from observations affected by shared errors. The use of estimates of the real covariance is recommended when the study objective is to evaluate or estimate the association between processes instead of the association between observations. A numerical example illustrates the bias in statistics and discusses the relevance of considering shared errors in linear regression and kriging.

Keywords

Sampling error segregation process sampling theory shared errors correlated errors correlation regression QAQC

Introduction

Geochemical, environmental, or mining databases commonly contain thousands of samples, each of which may have individual values for more than 50 different chemical elements or species. This abundance of data provides an opportunity to discover a wide range of geological processes within a surveyed area. Multivariate statistics makes it possible to interpret the significance and relationships of tens of variables. Inevitably, all sample measurements are affected by the intrinsic presence of sampling error (Gy 1982). Two or more variables may be affected by the same source of error. The error associated with observations of different variables may be highly correlated. The measured correlation or covariance from observations affected by this type of error can be greater than that of the underlying true process, overestimating the real association between geological processes based on data available.

Covariance and correlation are parameters required by almost all multivariate statistical methods. These statistics are directly affected by the proportion of shared errors in the total error. Ignoring shared errors and assuming the observed statistics are representative of the underlying true process may lead to wrong decisions. There are examples of methods directly impacted by this wrong assumption such as cluster analysis based on the structure of covariance within observations rather than on their means (Ieva et al. 2016), principal component analysis (Pearson 1901), or multivariate geostatistics (Wackernagel 2003).

The total error is divided into four components, shared multiplicative, shared additive, non-shared multiplicative, and non-shared additive errors. Shared errors arise from correlated characteristics of minerals that lead to similar sampling errors and biases on the observations of two or more variables. To our knowledge, no discussion exists in the geosciences related to how to model and infer the components of error or how to estimate the underlying true correlation and covariance from observations affected by shared errors. The developed model of errors in the present study employs the same assumption as the widely known Thompson–Howarth model (Thompson and Howarth 1973, 1976, 1978). The term ‘sample’ has been a source of confusion between the statistical and the geoscience literature. The first usually refers to samples as a set of observations from a population. However, in geoscience literature, all specimens of soil, rocks, stream sediments, etc. are often called ‘samples’. Within this paper, collected specimens are referred to as samples, while each measurement of a given variable of interest at this sample is referred to as an ‘observation’.

The aim of this study is to understand the presence of shared errors and propose an approach to estimate statistics between underlying true processes from observations affected by this type of error. This article is organised as follows: (1) it starts with a section where we study the presence of shared errors and discuss very common situations in which they may occur; (2) next, the classic model of errors is presented and extended to a novel multivariate model to deal with shared errors; (3) the issue of estimating the variance, covariance, and correlation of the underlying true processes from observations is addressed; (4) a solution using pairs of duplicates to estimate the four different types of errors is proposed; (5) a numerical experiment is used to illustrate how bivariate statistics measured from observations with shared errors may diverge from the real statistics; and (6) the paper concludes with examples of applications to which ignoring the presence of shared errors may lead to wrong decisions.

The occurrence of shared errors

Shared errors may arise in a variety of ways in very common geological situations. Suppose there are N variables of interest, k = 1, … , N, and the same sample is assayed to obtain their readings. Some sources of error may be shared within the analysed variables. Figure 1 illustrates four different levels of association between two variables. In Figure 1(a), both variables are associated with the same mineral assemblage structure, so their sampling errors are shared at any scale and granulometry. Figure 1(b–d) illustrates other levels of physical association.

The heterogeneity of a lot may be greatly affected by the relationship between physical properties of its minerals, leading to a state in which segregation or distribution heterogeneity is the rule, rather than the exception. Therefore, the only method to reduce the segregation factor is to homogenise the material before sampling (Pitard 1993). Figure 2 shows how an incorrectly taken sample may show an association that does not exist in the true geological process. The figure shows the initial homogeneity state (Figure 2(a)) of two different flake minerals (red and green) and the same minerals in a highly segregated state (Figure 2(b)). Three samples were taken from the non-homogenised lot (dashed-line areas in Figure 2(b)). The segregation process leads to a condition in which red and green minerals are simultaneously above or below the average proportion of the lot, creating a false correlation.

There are many other situations where shared errors occur. These are mainly related to variables associated with minerals that are affected in similar ways in a given segregation process during delimitation, extraction, and preparation steps. Examples of mineral differences that can simultaneously segregate two or more minerals from gangue minerals are (1) different shape and density; (2) magnetic properties; (3) electrostatic properties of liberated constituents occurring as tiny flakes, such as biotite and scheelite; (4) moisture content and/or adsorbing water capability; and (5) minerals adhering differently to the walls of a storage bin or having different angles of repose, etc. (Pitard 1993).

The true association between variables with shared errors may only be correctly estimated if the error model used considers this type of error. The next section presents the classic model of uncorrelated errors to be extended to the multivariate case.

The classic observation model

Observations z_i_,obs are indirect measurements of the underlying true process of Z_true. Each individual observation i, i = 1, … .,I, comes from z_i_,obs = z_true + ε_i, where ε_i in the simplest case is unbiased E{ε_i} = 0, uncorrelated with the random variable Cov{Z_i,_true, ε_i} = 0, and with a known variance Var{ε_i} = σ² _i _,error that may be different for each sample.

When there is a wide range interval in the observations, the sampling error can vary significantly over the range of grades, so the standard deviation alone cannot properly describe the error behaviour and its precision. In this case, the well-established fact that the average error increases as the grade increase needs to be included (Howarth and Thompson 1976; Thompson and Howarth 1973, 1976, 1978). Thompson and Howarth assumed unbiased data affected by normally distributed errors. The error is part of a function of the grade and part independent. In the resulting model, the unknown underlying true value is affected by additive and multiplicative errors: (1) where z_true: underlying error-free true; z_i_,obs: observation i of the underlying error-free true; : Sampling error variance associated with the observation z_i_,obs; X_m: Standard normal Gaussian values N(0, 1); A_k: magnitude of additive error component; C_k: magnitude of multiplicative error component.

The scatter diagram between matched pairs of duplicate samples should help to decide which model is more adequate based on the shape of the cloud of points. Figure 3 (green dots) shows the effect of additive errors. When C_k = 0 [Equation (1)], the scatter plot between duplicates shows a constant distance around the x = y line. In the case of multiplicative error (yellow dots), A_k = 0 (Equation (1)), the closer the grade is to zero, the smaller the error. Real datasets are commonly affected by both types of errors.

Equation (1) deals with situations where the errors of each variable are independent of the error attached to observations of other variables. We now develop a generalised model to deal with additive or multiplicative errors shared by two or more variables.

The model of observations in the presence of shared errors

The development of the generalised equation is based on adding two new components to Equation (1). (1) The parameter B is the magnitude of additive errors, and (2) D is the magnitude of multiplicative errors. These new components are shared by two or more variables k, affecting them with the same magnitude and proportional deviation between the observed and real value. The resulting model, where the total grade-proportional error is split into shared (D) and non-shared (C_k) and independent errors are divided into shared (B) and non-shared (A_k), is: (2) Equation (2) is a generalisation that embraces several cases, such as observations affected only by additive (C_k = D = 0) or multiplicative errors (A_k = B = 0), or combinations of them. Equation (1) model is a specific case where the shared error is null (A_k = C_k = 0). After defining the generalised model, we discuss how to infer the real statistics between the error-free variables from observations in the next section.

Inferring the geological structure of a correlation

The variance of Equation (2) combines the sampling and analytical errors plus the variance of the underlying true process: (3) can be estimated from observations in a given domain based on bilinearity properties. The non-shared additive A_k and multiplicative terms C_k are cancelled and the measured covariance is only affected by shared errors: (4) The correlation between error-free processes is the covariance between two error-free variables divided by the product of their standard deviations. Equations (3) and (4) become: (5) The standard deviation can be estimated by the observed variance minus the sampling error variance . Figure 4(a) shows observations between two error-free variables with a correlation r² = 0.63. The variables were perturbed by a total error C_k + D = 0.1 with D = 0.01, 0.05, and 0.09. The measured r² after adding the error is, respectively, 0.23, 0.51, and 0.77. It is interesting to highlight that the measured correlation between variables with shared errors of 90% is larger than the error-free correlation.

We present how to use pairs of duplicates to estimate the four different types of errors in the next step.

Estimating the error components

An exhaustive discussion about approaches to estimate the presence of shared errors is beyond the scope of this paper. However, some insights about how to infer A_k, B, C_k, and D using duplicate samples are discussed. Duplicates are used to control data quality and process repeatability. The duplicate must be collected at a particular stage of the sampling protocol, such as comminution and sub-sampling, then prepared and assayed following the same rules used for collecting the original sample. Matched pairs of chip samples or pairs of parts of a drill core are the best data to evaluate the combined contribution of all sources of errors, allowing monitoring the variance added by each subsequent stage.

Additive components A_k and B

We can find the value of by computing the variance of a set of measurements of samples with a null grade, which is possible because grade-proportional components are null, and A_k and B control all the variance. (6) At first glance, the best way to measure the null-grade variance is by measuring the variance of blank samples. QC routines commonly use materials without the grade of interest (e.g. barren quartz) instead of matrix-matched blanks. Non-matrix-matched blanks are cheaper and adequate to evaluate laboratory contamination and verify the correct sample handling. However, the variance of non-matched blanks is inadequate in the proposed approach because the rock matrix plays an important role in the error behaviour and variance.

Multiplicative components C_k and D

The linear regression slope is not affected by additive errors, so it can be used to infer the multiplicative components (C_k and D). Three different regressions allow us to estimate the values of , , and C₁ C₂:

The slope of the regression line between the variance and the average value of each matched pair is (Figure 5(a));

The regression between the difference Var{z_i,_{1, obs}} – Var{z_i,_{2, obs}} (computed with duplicates of each variable), and the difference of grades from the original observations has a slope of (Figure 5(b));

The regression between the covariance of measurements and the product of the grade of the original (or duplicate) samples of two observations i. The regression slope is C₁ C₂ (Figure 5(c)).

The three different results from the regression slope are used to define the individual values of each component by solving a linear system of these equations.

Simulation studies

A series of observations were simulated to illustrate how the four different types of error affect the measured variance, covariance, and correlation. A set of values X_i,_initial(i = 50, 000) were drawn from N(100, 10) and observations X_i,₁ and X_i,₂ were simulated from X_{i, k} = X_i,_initial + N(0, 5). The resulting variables had ρ(X_i,₁, X_i,₂) = 0.62, Cov{X_i,₁, X_i,₂} = 100, and

(Tables 1 and 2).

Table 1

Covariance and correlation between 50, 000 values drawn from N(100, 10) and perturbed individually by an error N(0, 5), resulting in ρ(X_i_,1, X_i_,2) = 0.62. Next, these values were perturbed by additive errors A_k + B = 10, with different proportions of shared (B) and non-shared (A_k) errors.

Real statistics		Observed statistics – additive error
Cov_real{X_i,₁, X_i,₂}		B	A_k	r²	Cov_obs{X_i,₁, X_i,₂}
100	11.2	0	10	0.19	100	14.2
100	11.2	1	9	0.24	102	14.2
100	11.2	3	7	0.36	109	14.2
100	11.2	5	5	0.52	126	14.2
100	11.2	7	3	0.66	147	14.2
100	11.2	9	1	0.76	180	14.2
100	11.2	10	0	0.79	200	14.2

Table 2

Covariance and correlation between 50, 000 values drawn from N_i(100, 10) and perturbed individually by an error N(0, 5), resulting in ρ(X_i_,1, X_i_,2) = 0.62. Next, these values are perturbed by multiplicative errors C_k + D = 0.1, with different proportions of shared (D) and non-shared (C_k) errors.

Real statistics		Observed statistics – multiplicative error
Cov_real{X_i,₁, X_i,₂}		D	C_k	r²	Cov_obs{X_i,₁, X_i,₂}
100	11.2	0	0.1	0.20	100	14.2
100	11.2	0.01	0.09	0.23	100	14.2
100	11.2	0.03	0.07	0.35	109	14.2
100	11.2	0.05	0.05	0.51	125	14.2
100	11.2	0.07	0.03	0.67	151	14.2
100	11.2	0.09	0.01	0.77	184	14.2
100	11.2	0.1	0	0.79	200	14.2

Next, the initial values were used to simulate observations X_i,₁ and X_i,₂, perturbed by additive or multiplicative errors. The total added error is constant to both cases, being C_k + D = 0.1 or A_k + B = 10. We varied the ratios between non-shared and shared components, C_k/D (Figure 4) and A_k/B, using 0, 10, 30, 50, 70, 90, and 100%. Tables 1 and 2 give the results of these simulations in the presence of additive or multiplicative error, respectively.

The non-shared components A_k and C_k do not change the measured covariance, and the higher the error, the lower the measured correlation between variables (Tables 1 and 2). Shared errors have an inverse impact; the bigger the error variance, the closer ρ(X_i,₁, X_i,₂) is to one since the measured correlation combines the underlying error-free correlation plus the shared-error correlation. In both cases, overlooking the type of error attached to observations can lead to statistics being non-representative of the geological processes of interest.

Applications

The true statistics cannot be directly measured from observations in real-world problems because available data are always affected by errors. Bivariate statistics naively inferred from these observations measure the association between observations but not the variance, covariance, and correlation between the underlying true processes of interest. Two situations in geosciences in which the covariance plays a relevant role in parameter estimation were used to show how the true statistics may be estimated using the developed equations and what needs to be done after we measure shared and non-shared errors and estimate the correct associations between variables Extending the proposed solution to other statistical applications is straightforward.

Linear regression in the presence of shared errors: In a linear regression y = α + Xβ, the ordinary least-squares (OLS) estimates of the intercept α and slope β are obtained from observations X. When observations are error-free, the OLS estimate of the regression slope, , is: (7) In the presence of errors that are non-shared and independent of grade, the variance of these errors sum with Var{X_true} (Equation (3)) while it does not affect Cov{Y_true, X_true} (Equation (4), Example I). The presence of errors only affects the denominator, leading to the well-known attenuation effect of the regression (e.g. Fuller 1987). Many variations of the original estimator are available to deal with errors that are non-shared and independent of grade (Kelly 2007).

A simple method of accounting for shared errors between X and Y in linear regression is a direct application of the proposed method, where the equation's slope is ‘debiased’ by replacing the statistics adjusted through available observations with the correct variance (Equation (3)) and covariance (Equation (4)). Next, we use these estimated moments to estimate the regression parameters of Equation (7): (8) From Equations (3) and (4), if the measurement error correlation has the same sign as the real association between X and Y, then the shared-error components (B and D) overestimate the observed covariance; otherwise, these components lead to underestimated covariances. This problem and source of bias are not considered when shared errors are ignored since non-shared components A_k and C_k do not change the measured covariance.

Collocated co-kriging in the presence of shared errors: Collocated co-kriging is commonly used when primary data are present in sparsely distributed points while secondary data are located at all points of the grid being estimated. To avoid matrix instability, collocated co-kriging makes use of the secondary data only at the nodes to be estimated. This solution relies on the Markov-type screening hypothesis, which assumes that these collocated data screen further away data of the same type (Xu et al. 1992). As a practical result of this assumption, we do not need to compute the cross-covariogram between primary and secondary data C₁₂(h). It is estimated from the direct covariogram of the primary variable C₁₁(h) and from the correlation coefficient between primary and secondary data C₁₂(0) (9) The collocated correlation structure is used to define kriging weights that minimise redundancy and maximise predictive ability. When C₁₂(0) is high, estimates are done based on secondary information. Conversely, estimated results are identical to the simple or ordinary kriging when C₁₂(0) = 0. In general, the lower ρ₁₂(0) is, the lower weights assigned to secondary data. Rocha et al. (2012) experimentally defined ρ₁₂(0) > 0.4 as the minimum correlation in which the use of secondary data is worthy.

In real-world problems, the unknown correlation between secondary data and the underlying true process of interest is assumed to be equal to the correlation between the association ρ₁₂(0) of secondary data and the process to be estimated ρ_{true, 2}(0). However, the use of secondary data must be approached with caution when in the presence of shared errors because constraining the kriging system to deal with an incorrect covariance structure and correlation model leads to weights that will not minimise the estimation error.

ρ_min is the minimal correlation in which the use of secondary data is worthwhile in estimates. In a case of overestimated correlation, ρ₁₂(0) > |ρ_min| and ρ_{true, 2}(0) = 0, and the larger the kriging weight attributed to the collocated-secondary data, resulting in larger estimation error. On the other hand, when ρ₁₂(0) = 0 and ρ_{true, 2}(0) > |ρ_min|, a set of secondary data that would help to minimise the estimation error may be discarded by practitioners or almost ignored in the kriging system. In both cases listed above, the decision of whether to use the secondary data and the correlation to be input into the kriging system should be defined by Equations (3) and (4). These equations measure the correct association between observations and the process to be estimated, leading to optimal kriging weights.

Conclusions

Shared errors may arise from the delimitation, preparation, and sub-sampling steps of a sampling protocol, creating a false correlation that is not present in the geological process. This paper proposes a general model that deals with four types of error, shared multiplicative and additive errors and non-shared multiplicative and additive errors. Next, the effect of each type of error in measured bivariate statistics was investigated. A long list of common situations where shared errors can arise during sampling was presented.

There are many applications where the correct covariance and correlation to be considered should measure the real association between the processes, not the association between observations. Constraining models to the correct statistics are important for the greatest prediction accuracy and precision. Our main finding was that the measured correlation and covariance may be substantially under or overestimated by overlooking the impact of the shared errors. When the magnitude of each type of error is known, it is possible to estimate the correct statistics from observations. A method to estimate these magnitudes using quality control data was presented.

Footnotes

Disclosure statement

No potential conflict of interest was reported by the author(s).

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Sampling error correlated among observations: Origin,impacts,and solutions

Abstract

Keywords

Introduction

The occurrence of shared errors

The classic observation model

The model of observations in the presence of shared errors

Inferring the geological structure of a correlation

Estimating the error components

Additive components Ak and B

Multiplicative components Ck and D

Simulation studies

Applications

Conclusions

Footnotes

Disclosure statement

References

Additive components A_k and B

Multiplicative components C_k and D